Self-stabilization of twin beams by a passive Kerr medium

1 May 1994

OPTICS COMMUNICATIONS ELSEVIER

Optics Communications 107 (1994) 420--424

Self-stabilization of twin beams by a passive Kerr medium C. F a b r e Laboratoire de Spectroscopie Hertzienne de FENS t, Universit~ P. et M. Curie, Tour 12, Case 74, 75252 Paris Cedex 05, France

R.J. Horowiez Depto. Fisica, Pontificia Universidade Catolica, Rio de Janeiro, R.J, Brazil

Received 21 October 1993

Abstract

A passive optical system is proposed which is able to produce an intensity squeezed light starting from twin beams having intensity correlated beams at the quantum level. It consists in using a nonlinear medium presenting a crossed Kerr effect inserted in an optical cavity. The intensity fluctuations of One beam modify the resonance condition of the cavity for the other, and therefore its intensity. It is shown that this system is able to produce a substantial noise reduction. 1. Introduction Twin beams are light beams which have the interesting property of presenting strictly identical intensities: they have equal mean intensities, but also identical quantum intensity fluctuations. One way to produce a good approximation of such twin beams is by use of an optical parametric oscillator (OPO) [ 1 ]. In such a system, a medium with a second order susceptibility X~2) is inserted in an optical cavity and pumped by an external light source of frequency 09o. Under appropriate phase-matching and cavity resonance conditions, two modes with frequencies 09~ and 092 are generated, which obey the energy conservation condition ~ot+co2=Wo. One can show that the quantum intensity fluctuations of the signal and idler modes are perfectly correlated for noise frequencies smaller than the cavity cut-off, provided that the losses of the system are very small [ 2 ]. This property Unit6 de recherche de l'Eeole Normale Sul~rieure et de l'Universit6 Pierre et Marie Curie, assoei~eau CNRS.

has been experimentally checked either in the cw [ 3,4 ] or the pulsed regime [ 5 ]. The experiments have shown that the difference between the intensity fluctuations of the signal and idler modes is reduced with respect to the standard quantum limit, which is in this problem the shot noise corresponding to the sum of the intensity of the twin beams. Because of the high optical quality and low absorption losses of available nonlinear crystals such as KTP, very large values for the noise reduction have been recently measured (86%), corresponding to actual correlations between the twin beams of better than 92% [ 6 ]. It is then possible to transfer the quantum correlation between the twin modes into a quantum noise reduction of a single mode. For example, the twin beams generated by an OPO have been used in an active feed-forward system which measures the intensity fluctuations of one twin and correct the intensity fluctuations of the other by means of an electrooptic modulator [ 7,8 ]. In this paper we propose another method for stabilizing the intensity of one of the two twin beams

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C. Fabre, R.J. Horowicz ~Optics Communications 107 (1994) 420-424

which uses a passive medium instead of an optoelectronics network. The basic idea is closely related to that proposed by Monken and Barbosa [9] for a semiconductor, and uses the fact that a Kerr medium, having a refractive index sensitive to the intensity of one mode, may tune an optical cavity into resonance with the other mode and therefore change its intensity. Kerr media in resonant cavities are well known to generate squeezed light when one injects a single mode in the cavity [ 10-13 ]. We show in this paper that a crossed Kerr effect can be used to transfer two-mode intensity correlation into one mode squeezing. The same crossed Kerr effect has been studied in Ref. [ 14 ] for the generation of squeezed beams and quantum correlated beams when one starts from uncorrelated shot noise limited input beams. It has also been experimentally used to transfer intensity fluctuations from one beam to another, thus providing an efficient Q N D device [ 15 ].

2. Outline of the method Our proposed system is shown in Fig. 1, the twin beams generated by an OPO are incident on a ring cavity, which contains the Kerr medium, through a coupling mirror having a high reflection coefficient for the signal and idler modes (rj will be the amplitude reflection coefficient of the mirror for modes j = 1,2 ). In a crossed Kerr medium, the refractive index n of the medium at the frequency of one mode is sensitive to the field intensity of the other mode through the crossed phase-modulation effect, and for simplicity we will neglect the self phase-modulation effect of each mode on itself (which leads to single

~ OPO~ nl

~ ~ ~

421

mode quantum noise reduction effects discussed in Ref. [ 14]). Such a configuration can be found for example in atomic systems when 0)~ + 0)2 is quasi-resonant with a two-photon transition [ 14 ]. In that case we can simply write n( 0)l ) = n°( 0)l ) + n'l I2 ,

n (0)2)--n°(0)2) + n'211,

(I )

where n 5 (j= I, 2) is the nonlinear crossed Kerr coefficient. Eq. ( l ) means that any intensityfluctuationin one of the incoming beams will affect the optical length seen by the other, and therefore will modify its resonance conditions. Let us now suppose that the cavity is resonant for, say, mode I, and presents some detuning for mode 2. If the incident intensity I~ of beam I decreases, the resonant condition for mode 2 will be modified by the Kerr effect and hence, for a convenient choice of the cavity length, the outgoing intensity i~ut will increase. Since beams 1 and 2 are twin beams, their fluctuations are identical and, as a consequence, the fluctuations of the incoming beam intensity I ~ will be compensated: a self-regulated beam with intensity/~ut will be produced at the output. As the twin beams are correlated at the quantum level, this stabilization scheme is able to reduce the intensity fluctuations of the outgoing field below the short-noise level. This intuitive reasoning allows us to understand why a noise reduction could be expected for that system. It does not however take into account the phase fluctuations of the incoming beam. Indeed, these fluctuations will be converted into intensity fluctuations by the optical cavity, and the behaviour of the outgoing beam will depend on both the phase and intensity fluctuations of the incoming beams. In what follows we will derive an expression for this dependence, and it will be shown that under appropriated conditions the system is indeed capable of generating intensity squeezed light.

3. Evolution equations for the fields Fig. 1. The twin beams a~ and ~x~*generated by an optical parametrie oscillator (OPO) enter a one port ring-cavitywhich contains a nonlinear medium with a cross Kerr effect. In some conditions, the output beam otl~t may be squeezed due to the intracavity cross Kerr effect between fields ~xland ~x2.

The time evolution of the classical field amplitudes aj U = 1,2) of the two modes inside the cavity obeys an equation which may be obtained by considering

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C. Fabre,R.J. Horowicz/ Optics Communications 107 (1994)420-424

the total phase acquired by the field in one round-trip time z through the optical cavity of length L. This phase will be given by the sum of the empty-cavity propagation phase 9 j r toyL/c and the Kerr-effect induced phase-shift Oj=Kj/h where It= Iotll 2 (1= 1, 2, l # j ) is the circulating field intensity of the other mode. The constant Kycan be related to the nonlinear coefficient n~ introduced in Eq. ( 1 ) by

Kj=n~tojL'/c ,

(2)

where L' is the length of the Kerr medium. In the parametric limit, it can be shown that KI =/(2= K, and for a sodium atomic beam of density 1012 at/era 3, an interaction length of 2 cm, a two-photon transition oh +oJ2 dctuned from resonance by 12 GHz, K is of the order of 7 × 10 -s cm2/W, ensuring a nonlinear phaseshift of 0.01 for intracavity powers of 22 mW focused on an area of 1.6 × 10 -4 cm 2 [ 16 ]. In the good-cavity limit, and for a cavity which is quasi-resouant for both modes, the time evolution equations read T(~I = -- (?1 + i ~ 1 "~'iKI 0/2 [ 2)0/1 "~ ~ 1 0 / ~

n,

z&2 = - (72 + i 5~2 +iKI a112)a2 + x / ~ 2 a ~ n ,

(3)

where ?j= 1 -ry~t~ 1, 8~j=q~y-2pjn~ 1, (py integer) and ~t)~ is the amplitude of input fields incident on the cavity, which we assume to be equal to the fields generated by the OPO, as calculated for example in Ref. [2]. Let us first consider the mean values of the different fields. The twin incident fields are characterized by equal mean intensities, and we will take the corresponding amplitudes as real quantities: (¢t~)=(tx~)=(tx

~ ) (real).

(4)

tensity, and therefore the Kerr phaseshift for mode 2, is maximized. As for mode 2, its mean detuning should be properly chosen so that a decrease in the intensity of mode 1, and hence of the optical length, would bring it closer to resonance. This means that the empty cavity should be placed in the negativeslope part of the Fabry-P~rot response curve for mode 2. In that case, under appropriate conditions, we expect beam 2 to be stabilized at the output. In the following, it will be useful to introduce the real variables pj and qj describing, respectively, the real and imaginary parts of the field amplitudes: (6)

Olj -~ (pj + iqj ) / X/2 .

In the stationary state, we find from Eqs. (3) the steady-state solution for the mean values (P2) and ( q2) of the quadrature variables P2 and q2: 2?2/2 (P2> = ~2.t.q~2 (oLin> ,

2y1/2~ ------ ~2.i.'-""" ~ (t~in> •

(7) To determine the quantum fluctuations of the two fields, we will use the semiclassical input-output method, in which the quantum fields are treated as stochastic variables obeying the classical equation of motion and having input statistical distributions mimicking the quantum statistical distribution of the pump fields and of the vacuum fields. This approach gives the correct answer for the variances of the output fields in the small quantum fluctuation limit, i.e. when it is possible to linearize the equation of motion (3) around the mean values (7). We therefore write

OLj(t ) = ( Olj> + iotj( t ) ,

(8)

We then assume that, in presence of the intracavity mean fields, the mode 1 is resonant with the cavity, whereas the mode 2 presents some total detuning 802:

where the field fluctuations 8aj are small quantities with zero mean values and linearize Eqs. (3) around the steady state. One gets

8~1 + K < I 2 ) = 0 ,

• s~l=-?~ 8pl + v / ~ sp~~,

8~2 +K(I~ > = ¢ ~ 0 .

(5)

The resonance condition of mode 1 (first Eq. (5)) implies that the phase fluctuations of the incident field a p will not be converted into intensity noise by the cavity, and hence will not contribute to the intensity noise of the circulating field or2. Furthermore, it ensures the dynamical stability of the system [ 14 ] and corresponds to a situation in which the circulating in-

z 5/)2 =--?2 5p2+q~Sq2+K(q2> 811 + x ~ 2

5p~,

T 5q2 -~ --72 8q2 --q~ 8P2 - K ( p 2 ~ 511 + x / ~ 2 5 q ~ ,

(9) where 8p~ and 8q~ are the real and imaginary parts of the fluctuations associated to the incident field,

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C. Fabre, R.J. Horowicz / Optics Communications 107 (1994) 420-424

coming from the OPO. As we have chosen the incident fields as real quantities, the fluctuations ~p)n, fiqjm correspond respectively to the phase and intensity fluctuations of the incident fields. Eq. (9) shows that in absence of the Kerr medium (K=0), the two modes are uncoupled, as could be expected, whereas by inserting the medium the fluctuations of mode 2 will be affected by the intensity fluctuations 811 of mode 1. These intensity fluctuations may be expressed in terms of the quadrature fluctuations 8pl, 8ql by using Eq. (6): 811 =

5pl + (ql > 8ql.

(10)

We are interested in the noise of the field detected outside of the cavity, and therefore we will use the boundary conditions imposed by the coupling mirror to express the outgoing field amplitude ot°ut in terms of the incident field ot~ and the circulating fields c~. In the good-cavity limit yj,~ 1 these conditions read: oq°at ---x/-~j o+- or)".

(11)

By taking the Fourier transform of the time evolution equations (9), we reduce them to algebraic equa= tions, which may be solved for the Fourier compo= nents of the field fluctuations 8/~(D), 8~(D). Using the boundary conditions (I I ), we find for the output field fluctuations, in terms of the incident beam fluctuations: 1 8P2(D) = D ( D )

2y2 ¢K8~ (D) X(--2V'~r2LOX+ y2+ip2 +2721p 8(~n(~r~) -t- (.Q2T2"I"r2 _¢2) 8p~n(~Q)), 1

D ( D ) = (LQt+ Y2)2'+¢2,

(13)

2V/2 (otiS> 8#~(D).

5T~ ( D ) = iDz+ ?t

(14)

As noticed above, the output fluctuations depend on the correlated intensity fluctuations 8p~n and 8p~n of the input twin modes and on the phase fluctuations 8q~U of the incident mode 2, which are assumed to be the output fluctuations of the OPO. These fluctuations may be found in Refs. [ 2,17 ]. Introducing the coupled variables 8pm+=~p~+~p~2n,

~p~=Sp~--Spi~ n ,

(15)

one has, for an OPO without extra-losses and exactly tuned to resonance: F2

<18pm+(D)12>=l+ .Q2-I-F2 (0-_ 1 ) 2' ~r~2

< 18pP(~2)[2> = .Q2_I.F2, /.40. 2

< ]~q~n(.Q) 12> ----½ 1-t" 2.Q2(~--~F202)],

(16)

where F is the OPO cavity bandwidth and a is the OPO pump parameter (a= 1 at the OPO threshold). According to definition (6), one has, for a coherent state of vacuum: < [ 8p(~2)12> = < 18q(co)[2> = 1/2. Note that, at zero frequency D=O, the OPO beam phase noise is infinite, whereas the noise on the amplitude difference 8p~ (D) is zero. This is due to the phase diffusion phenomenon which occurs in a nondegenerate OPO [ 18 ]. So it is not possible to take full advantage of the twin character of the OPO beams, because of the presence of this infinite phase noise which is also coupled to the output field intensity noise.

~2(t2)- D(~)

X(2.V/~2 -2y2~

--i~T~2 Jt'lp2--72 KS~I (,,,~) },22 + @2

4. Intensity spectrum of the output field

,Sp#(a)+ (a2~-2+},2-~)') 8~#(a)), (12)

where

In order to find the intensity fluctuations of the outgoing field <82I~ut(D)>, we use Eqs. (6), (10) and (12). We find the following expressions for the outgoing intensity fluctuations of beam 2:

424

C. Fabre, R.£ Horowicz / Optics Communications 107 H994) 420-424



S ( ~

.

.

.

.

.

=

q

,

r

< [ 8l~ut(g2)[2> = 2(Y22 + ~ 2 ) 2 [ D ( - Q ) ]2

× { [ ( B + A ) 2 + C 2] < 8 2 p ~ > + [ ( B - - A ) 2 + C 2 ] < 18pP12> + 16722~4z 4 8 ~ 2 },

(17) 0

where A = (y2 .~_~2) 2 +~,-~2.c2(72 _ ~ 2 ) , B---~8~2T2{o~2K< ¢x in ) 2/(~'~ 2T2 + 72 ) ,

C=Sf~r~27tK( ain) 2/ (f~2z2 + 72) ,

(18)

and the input noise spectra are given by Eqs. (16). The shot noise level is given by < ] 8/~ut (~,'2) [2> shot= .

(19)

Let us consider the normalized spectrum S ( ~ ) = < 181( ~2) 12> / < [8 ( £2) 12) shotderived from Eqs. ( 17 ), (19). For ~ = 0 , S ( ~ ) is equal to [ 1 + (or- 1 ) - 2 ] / 2 , which is the input zero-frequency intensity noise of each OPO beam [ 17 ], left unchanged by the nonlinear interaction, like in the self phase modulation case [ 14]. One also finds that S(£2) = 1 when C2--,c0: for such high frequencies outside both Kerr-cavity and OPO-cavity bandwidth, the two intracavity devices are not able to modify the input noises which are at the vacuum level. The amount of squeezing obtained in this system depends on three parameters: the ratio 71/72 of the mirror transmission coefficients, the normalized Kerr phase A=K(lin>/7 2, and the normalized cavity detuning ~/Y2. For the sake of simplicity, we will restrict our analysis to the case a = 2 , when the incoming beams have then an intensity noise at the shot noise level ( ( [ S p ~ ( ~ ) 12> -- < 18 p ~ ( ~ ) 1 2 ) = 1/2). The best noise reduction is obtained when the OPO cavity linewidth is of the order of, or larger than, the Kerr cavity linewidth for mode 2, this latter being much larger than the Kerr cavity linewidth for mode 1. Fig. 2 shows the result obtained for a particular choice of parameters, compatible with experimental values. We can see that the output intensity noise spectrum presents a deep minimum for frequencies different from zero and smaller than 72/z. These results thus show that the proposed scheme may provide an almost perfect intensity squeezing.

1 Frequency ~r/3,2

2

Fig. 2. Spectrum of the output field intensity fluctuations S ( ~ ) ( S ( f l ) = 1 corresponds to the shot noise level). The frequency axis is normalized in units of Kerr cavity bandwidth for the squeezed mode. Normalized Kerr phase A=0.4, cavity detuning for field 2~ - - 0.372, transmission coefficient ratio yt/y~ = 0.01, ratio between OPO cavity linewidth and Kerr cavity linewidth for beam 2: 0.5, OPO pump parameters ~= 2.

Acknowledgements The authors are glad to acknowledge interesting discussions with Drs. C. Monken and P. Ribeiro. References [ 1 ] S. Reynand, C. Fabre and E. Giacobino, J. Opt. So¢. Am. B 4 (1987) 1520. [2 ] C. Fabre, E. Giacobino, A. Heidmann and S. Reynaud, J. de Physique 50 (1989) 1209. [3] A. Heidmann, R.J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre and G. Carny. Phys. Rev. Lett. 29 (1987) 2555. [4 ] C.D. Nabors and R.M. Shelby, Phys. Rev. A 42 (1990) 556. [ 5 ] O. Aytur and P. Kumar, Phys. Rev. Lett. 65 ( 1990 ) 1551. [ 6 ] J. Mertz, T. Debuisscbert, A. Heidmann, C. Fabre and E. Giacobino, Optics Lett. 16 ( 1991 ) 1234. [7 ] J. Mertz, A. Heidmann, C. Fabre, E. Giacobino and S. Reynaud, Phys. Rev. Letters 64 (1990) 2897. [8] J. Mertz, A. Heidmann and C. Fabre, Phys. Rev. A 44 ( 1991 ) 3229. [9]C. Monken and G. Barbosa, Phys. Rev. Lett. 67 (1991) 3372. [ 10] L. Lugiato and G. Strini, opticsComm. 41 (1982) 67. [ 11 ] M. Collett and D.F. Walls, Phys. Rev. A 32 (1985) 2887. [12] R.M. Shelby, M. Levenson, D.F. Walls, A. Aspect and G. Milburn, Phys. Rev. A 33 (1986) 4008. [ 13 ] S. Reynaud, C. Fabre, E. Giacobino and A. Heidmann, Phys. Rev. A 40 (1989) 1440. [ 14] P. Grangier and J.F. Roch, Quantum Optics 1 (1989) 17. [ 15 ] P. Grangier, J.F. Roch and G. Roger, Phys. Rev. Lett. 66 (1991) 1418. [ 16 ] J.F. Roch, thesis, University Paris Sud (1992) unpublished. [ 17 ] E. Giacobino, C. Fabre, S. Reynaud, A. Heidmann and R.

Horowicz, in: Photons and quantum fluctuations, eds. Pike and Walther (Adam-Hilger, 1988) p. 81. [ 18] J.Y. Courtois, A. Smith, C. Fabre and S. Reynaud, J. Mod. optics 38 (1991) 177.